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- CONTENTS 1987 - VOLUME 305 - SECTION I - N° 15
- .......... Page(s) .......... 665
- .......... Page(s) .......... 665
- We denote the binary predicate of coprimeness (x y meaning that x and y are coprime), and S the successor function (S, ). We show that there are (S, -definable functions f and g from N into the set of primes, such that if it is possible to (S, )-define for one among the f' s or the g' s, and for every positive integer x the sum x+f' (x) or the product xg (x), then the first order arithmetic is (S, )-definable.
- .......... Page(s) .......... 669
- .......... Page(s) .......... 669
- We compare the weak solution (
, h) of the Stefan problem in some annular domain , with the weak solution (, H) of the "symmetrized" problem, in , where is a symmetrical annulus having the same measure as . For the one-phase Stefan problem - when - we show, in particular, that the "volume of ice" (mes ) remains greatest in spherical symmetry (with initial data decreasing along the radii). - .......... Page(s) .......... 673
- We give a course of spectral results related to the positiveness of the transport semigroup. Necessary conditions and sufficient ones for the existence of eigenvalues are given. We present a new approach to the irreducibility of the transport semigroup. We exploit a result by I. Marek [1] to obtain strict monotonicity properties of the principal eigenvalue with respect to the different parameters of the transport equation.
- .......... Page(s) .......... 685
- .......... Page(s) .......... 685
- Two linear torsionless connections and D are naturally given on certain differentiable hypersurfaces ME of a reflexive Banach space E.
- The definition of these connections is based on the existence of a field P: ML (E; E) of "projections of minimal distance" as well as on the associated field N of "unit normals of Birkhoff-James", with Ker ().
- .......... Page(s) .......... 689
- .......... Page(s) .......... 689
- Frostman's theorem for trees and modulus [1] is applied to ultrametric Cantor sets and in general to regular Cantor sets. We get, for these sets, equality for the different dimensions usually considered.
- In this context we introduce the notion of presentation of a Cantor set. It is given by a tree and a modulus. Two examples enlightens the usefulness of this notion.
- .......... Page(s) .......... 693
- .......... Page(s) .......... 693
- Let B(x), , be a three parameters Levy Brownian motion. Denote by L(H+), L(H-) the closed subspaces of L2 spanned respectively by and .
- We prove that conditional on the processes B(x), and B(y), are independent and identical in law, moreover the associated covariance function coïncides with the Green function of the second kind for the bilaplacian defined on . Some consequences of this result are mentionned.
- .......... Page(s) .......... 697
- .......... Page(s) .......... 697
- To avoid any numerical locking in the finite element approximation of viscoelastic flow problems, we propose a three-field finite element approximation of this problem. This approximation, which involves velocities, stresses, and pressures is proved to converge for all times.
- .......... Page(s) .......... 701
- .......... Page(s) .......... 701
- We present a relaxed version of incompressible Euler equations, that permit foliated flows involving two velocities. These relaxed equations allow a two-phase evolution of some vortex sheets, as an alternative to discontinuous solutions of Euler equations.
- COMPTES RENDUS DE L'ACADEMIE DES SCIENCES MATHEMATIQUE 1987 - Tome 305 - Série I - n° 15